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Separable equation

A separable equation can be written so that the derivative is a product of something that depends only on $x$ and something that depends only on $y$ . You gather $y$ on one side and $x$ on the other, and integrate both. The constant is determined by an initial value. For $y' = 6x$ the right-hand side does not depend on $y$ , and integration gives a polynomial. Insert the starting point before you calculate a later value.

\int \frac{1}{g(y)}\,dy = \int f(x)\,dx

separation

y' = 6x \Rightarrow y = 3x^2 + C

a simple case

Symbols

$C$	the constant
$f,\ g$	the functions of one variable

Example

$y' = 6x$ and $y(1) = 4$ . Then $y = 3x^2 + C$ , and $3 + C = 4$ gives $C = 1$ . Therefore $y(2) = 12 + 1 = 13$ .

Integrate both sides before you insert the initial value. The constant comes from the integration.

Practise differential equations for free →

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Part of Mathematics R2: Differential equations.